Continuum mechanics has been used for simulating continuous matter such as solids and fluids (i.e., liquids and gases). Differential equations are employed in solving problems in continuum mechanics. Many numerical procedures have been used. One of the most popular methods is finite element analysis (FEA) or finite element method (FEM), which is a computerized method widely used in industry to model and solve engineering problems relating to complex systems such as three-dimensional non-linear structural design and analysis. FEA derives its name from the manner in which the geometry of the object under consideration is specified. With the advent of the modern digital computer, FEA has been implemented as FEA software. Basically, the FEA software is provided with a grid-based model of the geometric description and the associated material properties at each point within the model. In this model, the geometry of the system under analysis is represented by solids, shells and beams of various sizes, which are called elements. The vertices of the elements are referred to as nodes. The model is comprised of a finite number of elements, which are assigned a material name to associate with material properties. The model thus represents the physical space occupied by the object under analysis along with its immediate surroundings. The FEA software then refers to a table in which the properties (e.g., stress-strain constitutive equation, Young's modulus, Poisson's ratio, thermo-conductivity) of each material type are tabulated. Additionally, the conditions at the boundary of the object (i.e., loadings, physical constraints, etc.) are specified. In this fashion a model of the object and its environment is created.
Once the model is defined, FEA software can be used for performing a numerical simulation of the physical behavior under the specified loading or initial conditions. FEA software is used extensively in the automotive industry to simulate front and side impacts of automobiles, occupant dummies interacting with airbags, and the forming of body parts from sheet metal. Such simulations provide valuable insight to engineers who are able to improve the safety of automobiles and to bring new models to the market more quickly. The simulation is performed in time domain meaning the FEA is computed at many solution cycles starting from an initial solution cycle, at each subsequent solution cycle, the simulation time is incremented by a time step referred to as ΔT. Such simulation is referred to as time-marching simulation.
One of the most challenging FEA tasks is to simulate an impact event involving a structure undergoing very large deformation, for example, car crash or explosion simulations. As the modern computer improves, engineers not only wish to simulate the behavior in an impact event with structural failure, they also want to simulate structural behaviors after yielding before total failure from an impact event. However, it is difficult to simulate such phenomena with FEA using solid elements. For example, solid elements representing foam material of a bumper may be squeezed or compressed to become too distorted or squished thereby resulting into zero or negative volume, which causes numerical problem in the simulation (e.g., simulated aborted due to invalid number in a digital computer).
To solve the zero or negative volume problem, those failed solid elements are replaced with particles under smoothed particle hydrodynamics (SPH). However, mathematical formulations of the FEM and SPH are different. In order to have particles and solid elements coexist in the same of model, some kind of connections must be established to connect the particles and the solid elements. Prior art approach has been using a tied interface, which rigidly connects certain particles with solid elements. However, this approach generally leads to very unrealistic simulated results due to arbitrary placement of tied interfaces (i.e., rigid links). For example, particles and solid elements are tied together could be reasonable initially. But, as they deform in an unpredictable manner, arbitrary placement of these rigid links might result in a very unrealistic connections.
Therefore, it would be desirable to have a more realistic interfaces in a computer aided engineering analysis model such that SPH particles and FEM solids can coexist to avoid problems and shortcomings of the prior art approaches.